HOMEWORKS AND OTHER EXERCISES (will be)here.

Homeworks to hand in by the 19th of February: 1.3; 1.4; 1.7; 1.9; 1.10

No. of Credits: 3

No. of ECTS credits: 6

Time Period of the course: Winter Semester

Prerequisites: Probability 1

Course Level: PhD

Syllabus: will be here.

Classes: Tuesdays from 09:00 in room 301.

Course Coordinator: Imre Péter Tóth.

Planned schedule

week # | when | topic | remark |

week 1 | 2019.01.08 | Introduction, overview. Finite dimensional distributions and construction of stochastic processes. Kolmogorov extension theorem. | |

week 2 | 2019.01.15 | Classification of stochastic processes, examples. Poisson process, Wiener process. Markov property, finite Markov chains. | |

week 3 | 2019.01.22 | Mixing and ergodicity of Markov chains. Countable Markov chains. Transience, recurrence, positive recurrence. Random walks on Z^d, Pólya's theorem. | |

week 4 | 2019.01.29 | Random walks on Z. The reflection principle and applications. Distribution of the maximum, arcsine law, return times, local time. | CLASS DELAYED to the next three weeks |

week 5 | 2019.02.05 | Discrete time martingales. Branching processes. Barabási-Albert graph model, preferential attachment. | CLASS LONGER with 35 minutes + a break |

week 6 | 2019.02.12 | Discrete time stochastic integration. Discrete Black-Scholes formula. | CLASS LONGER with 35 minutes + a break |

week 7 | 2019.02.19 | Brownian motion, Wiener process. Construction(s), regularity properties, scaling properties. | CLASS LONGER with 35 minutes + a break |

week 8 | 2019.02.26 | Markov chains in continuous time. Infinitesimal generator, exponential semigroup. Examples. | |

week 9 | 2019.03.05 | Filtrations and stopping times. Markov processes and martingales in continuous time. | |

week 10 | 2019.03.12 | Ito's stochastic integral. Motivation, definition, basic properties. | |

week 11 | 2019.03.19 | Itô processes and the Itô formula. Examples of stochastic differential equations. | |

week 12 | 2019.03.26 | Itô representation theorem, Martingale representation theorem. |

Suggested literature:

- R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
- R. Durrett: Essentials of Stochastic Processes. 2nd edition, Springer, 2012.
- S: Karlin, H.M. Taylor: A First Course in Stochastic Processes. 2nd edition, Academic Press 1975.
- S: Karlin, H.M. Taylor: A Second Course in Stochastic Processes. Academic Press 1981.
- R. Durrett: Stochastic Calculus: A practical Introduction. CRC Press, 1996.
- F.C. Klebaner: Introduction to stochastic calculus with applications. 2nd edition, Imperial College Press, 2005.
- K.L. Chung, R.J. Williams: Introduction to stochastic integration. 2nd edition, Birkhauser 1990.
- D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. 3rd edition, Springer 1999.

There will be three homework assignments -- roughly once in every four weeks -- worth 20% of the total score. A midterm exam focused on problem solving will be worth 30%, while the final exam, focused on theory will be worth 50%.

Requirements for "audit":

regular participation in class and a short oral account of the concepts and phenomena learned.